Find the value of x if √5x⁴ + 3 = √5x.

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Points to Remember:

  • The equation involves square roots and variables.
  • We need to isolate ‘x’ to find its value.
  • Squaring both sides might be necessary to eliminate the square roots.
  • We need to check for extraneous solutions after solving.

Introduction:

This question requires finding the value of the variable ‘x’ in the equation √(5x⁴) + 3 = √(5x). This is a mathematical problem that necessitates the application of algebraic manipulation techniques to isolate and solve for ‘x’. Solving such equations often involves careful consideration of the domain of the variables to avoid extraneous solutions – solutions that arise from the algebraic process but do not satisfy the original equation.

Body:

1. Squaring Both Sides:

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To eliminate the square roots, we square both sides of the equation:

(√(5x⁴) + 3)² = (√(5x))²

This expands to:

5x⁴ + 6√(5x⁴) + 9 = 5x

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2. Isolating the Square Root Term:

Next, we isolate the term containing the remaining square root:

6√(5x⁴) = 5x – 5x⁴ – 9

3. Squaring Again:

Squaring both sides again to eliminate the remaining square root:

(6√(5x⁴))² = (5x – 5x⁴ – 9)²

This simplifies to:

180x⁴ = 25x² – 50x⁵ – 90x + 25x⁸ + 90x⁴ + 81

4. Rearranging into a Polynomial Equation:

Rearranging the equation into a standard polynomial form:

25x⁸ – 50x⁵ – 270x⁴ + 25x² – 90x + 81 = 0

This is a high-order polynomial equation. Solving this directly is complex and may require numerical methods or specialized software. However, let’s try a simpler approach by observing that if x=1, the original equation becomes:

√(5(1)⁴) + 3 = √(5(1))

√5 + 3 = √5

This is clearly false. Let’s try another approach.

5. Alternative Approach (Considering Potential Simplifications):

The original equation suggests a potential simplification if we assume x is positive. Let’s try to simplify the equation by factoring:

√(5x⁴) + 3 = √(5x)

If we assume x = 1/5, then:

√(5(1/5)⁴) + 3 = √(5(1/5))

√(1/125) + 3 = √1

This is also not true. The high-order polynomial equation obtained earlier makes it clear that a simple algebraic solution is unlikely. Numerical methods would be required to find the approximate solution(s) for x.

Conclusion:

The equation √(5x⁴) + 3 = √(5x) leads to a complex high-order polynomial equation (25x⁸ – 50x⁵ – 270x⁴ + 25x² – 90x + 81 = 0) which is difficult to solve algebraically. Simple substitution of values does not yield a straightforward solution. Therefore, numerical methods (such as the Newton-Raphson method or using software like Wolfram Alpha) are necessary to find approximate solutions for x. It’s crucial to verify any solutions obtained numerically by substituting them back into the original equation to ensure they are not extraneous solutions. The focus should be on employing appropriate numerical techniques to solve this type of equation effectively. Further exploration using numerical analysis techniques is recommended to find the solution(s) for x.

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